
where we have used \(u\ge 0\)  in \(B_R\)  and \(w_\ell :=\bar{u}^\frac{\ell -m}{\ell }\) . Next, for the first integral \(I_1(p)\) , from [1]}, we have
\(I_1(p)\le -c \int _{ B_{r}}\int _{ B_{r}} |w_p(x)-w_p(y)|^p\phi ^p(y) d\mu _p+ c\int _{ B_{r}}\int _{ B_{r}}|\phi (x)-\phi (y)|^p(w_p(x)^p+w_p(y)^p)d\mu _p.\) 
